Progresses and challenges in forward prediction, Bayesian inversion and data integration 

Computer models, such as numerical solutions of PDEs are widely used in forward prediction, whereas the computing cost prohibits the use of computer experiments for some large-scale systems. In this talk, we will review the Gaussian process emulator for approximating computer models, and introduce new results on approximating functions with massive coordinates, high dimensional input and functional. Applications include emulating the TITAN2D model of pyroclastic flows, ground deformation simulation by COMSOL Multiphysics, and ab initio molecular dynamics simulations by density functional theory. For Gaussian processes with large observations, we will discuss marginalization of latent states. As an example, we will introduce stochastic differential equation representation of Gaussian process with Matern covariance for 1D input, and compute it in linear operations with respect to the number of observations, as an exact, computationally efficient alternative. We further introduce a new method, called generalized probabilistic principal component analysis, for matrix observations (such as images with irregular missing values) based on SDE representation for massive observations. Finally, Bayesian inversion will be discussed with applications on estimating system-specific information using microscopic video and satellite radar interferogram.